Mass–energy equivalence

For closed systems made up of many parts, like an atomic nucleus, planet, or star, the relativistic energy is given by the sum of the relativistic energies of each of the parts, because energies are additive in these systems. If a system is bound by attractive forces, and the energy gained in excess of the work done is removed from the system, then mass is lost with this removed energy. The mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up.^[15] This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons. This effect can be understood by looking at the potential energy of the individual components. The individual particles have a force attracting them together, and forcing them apart increases the potential energy of the particles in the same way that lifting an object up on earth does. This energy is equal to the work required to split the particles apart. The mass of the Solar System is slightly less than the sum of its individual masses.

For an isolated system of particles moving in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers, even those in relative motion. It is defined as the total energy (divided by c²) in the

The prediction that all forms of energy interact gravitationally has been subject to experimental tests. One of the first observations testing this prediction, called the Eddington experiment, was made during the Solar eclipse of May 29, 1919.^[16][17] During the solar eclipse, the English astronomer and physicist Arthur Eddington observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational attraction of light by the Sun. The observation confirmed that the energy carried by light indeed is equivalent to a gravitational mass. Another seminal experiment, the Pound–Rebka experiment, was performed in 1960.^[18] In this test a beam of light was emitted from the top of a tower and detected at the bottom. The frequency of the light detected was higher than the light emitted. This result confirms that the energy of photons increases when they fall in the gravitational field of the Earth. The energy, and therefore the gravitational mass, of photons is proportional to their frequency as stated by the Planck's relation.

Efficiency

In some reactions, matter particles can be destroyed and their associated energy released to the environment as other forms of energy, such as light and heat.

Unlike a system's energy in an inertial frame, the relativistic energy (${\displaystyle E_{\rm {rel}}}$) of a system depends on both the rest mass (${\displaystyle m_{0}}$) and the total momentum of the system. The extension of Einstein's equation to these systems is given by:^{[26][27][note 2]}

${\displaystyle {\begin{aligned}E_{\rm {rel}}^{2}-|\mathbf {p} |^{2}c^{2}&=m_{0}^{2}c^{4}\\E_{\rm {rel}}^{2}-(pc)^{2}&=(m_{0}c^{2})^{2}\end{aligned}}}$

or

${\displaystyle {\begin{aligned}E_{\rm {rel}}={\sqrt {(m_{0}c^{2})^{2}+(pc)^{2}}}\,\!\end{aligned}}}$

where the ${\displaystyle (pc)^{2}}$ term represents the square of the

and reduces to ${\displaystyle E_{\rm {rel}}=mc^{2}}$ when the momentum term is zero. For photons where ${\displaystyle m_{0}=0}$, the equation reduces to ${\displaystyle E_{\rm {rel}}=pc}$.

Low-speed expansion

Using the Lorentz factor, γ, the energy–momentum can be rewritten as E = γmc² and expanded as a power series:

${\displaystyle E=m_{0}c^{2}\left[1+{\frac {1}{2}}\left({\frac {v}{c}}\right)^{2}+{\frac {3}{8}}\left({\frac {v}{c}}\right)^{4}+{\frac {5}{16}}\left({\frac {v}{c}}\right)^{6}+\ldots \right].}$

For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because v/c is small. For low speeds, all but the first two terms can be ignored:

${\displaystyle E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}.}$

In classical mechanics, both the m₀c² term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured and so the m₀c² term is ignored in classical physics. While the higher-order terms become important at higher speeds, the Newtonian equation is a highly accurate low-speed approximation; adding in the third term yields:

${\displaystyle E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}\left(1+{\frac {3v^{2}}{4c^{2}}}\right)}$.

The difference between the two approximations is given by ${\displaystyle {\tfrac {3v^{2}}{4c^{2}}}}$, a number very small for everyday objects. In 2018 NASA announced the Parker Solar Probe was the fastest ever, with a speed of 153,454 miles per hour (68,600 m/s).^[28] The difference between the approximations for the Parker Solar Probe in 2018 is ${\displaystyle {\tfrac {3v^{2}}{4c^{2}}}\approx 3.9\times 10^{-8}}$, which accounts for an energy correction of four parts per hundred million. The

${\displaystyle 2.2\times 10^{-5}}$

Applications

Application to nuclear physics

The

A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by c²), which was given off as heat when the molecule formed (this heat had mass). Similarly, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion; in this case the mass difference is the energy and heat that is released when the dynamite explodes. Such a change in mass may only happen when the system is open, and the energy and mass are allowed to escape. Thus, if a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation.^{[note 3]} Thus, a 21.5 kiloton (9×10¹³ joule) nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If a transparent window passing only electromagnetic radiation were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass that absorbed the X-rays (and other "heat") would gain this gram of mass from the resulting heating, thus, in this case, the mass "loss" would represent merely its relocation.

Practical examples

Einstein used the

• 89.9 
  petajoules
• 25.0 billion
  GW·h
  )
• 21.5 trillion kilocalories (≈ 21 Pcal)^{[note 4]}
• 85.2 trillion BTUs^{[note 4]}
• 0.0852 quads

or the energy released by combustion of the following:

• 21 500 
  kilotons of TNT-equivalent energy (≈ 21 Mt)^{[note 4]}
• 2630000000 litres or 695000000 US gallons of automotive gasoline

Any time energy is released, the process can be evaluated from an E = mc² perspective. For instance, the "

• A spring's mass increases whenever it is put into compression or tension. Its mass increase arises from the increased potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring.
• Raising the temperature of an object (increasing its thermal energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum and iridium. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = 1×10⁻¹² g).^{[note 5]}
• A spinning ball has greater mass than when it is not spinning. Its increase of mass is exactly the equivalent of the mass of
  the Earth itself is more massive due to its rotation, than it would be with no rotation. The rotational energy of the Earth is greater than 10²⁴ Joules, which is over 10⁷ kg.^[37]

While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass, though nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields.^[38][39][40] Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.^[41][42]

Developments prior to Einstein

Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?"^[43] Swedish scientist and theologian Emanuel Swedenborg, in his Principia of 1734 theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it.^[44][45]

During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various

In 1905, independently of Einstein, French polymath Gustave Le Bon speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics.^[55][56]

Electromagnetic mass

There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists

Another way of deriving a type of electromagnetic mass was based on the concept of radiation pressure. In 1900, French polymath Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass^[4]

${\displaystyle m_{em}={\frac {E_{em}}{c^{2}}}\,.}$

By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.^[40]

Austrian physicist

${\displaystyle m_{0}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}}$

to the cavity's mass. He argued that this implies mass dependence on temperature as well.^[58]

Einstein: mass–energy equivalence

Einstein did not write the exact formula E = mc² in his 1905

In developing special relativity, Einstein found that the kinetic energy of a moving body is

${\displaystyle E_{k}=m_{0}c^{2}(\gamma -1)=m_{0}c^{2}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right),}$

with v the velocity, m₀ the rest mass, and γ the Lorentz factor.

He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the correspondence principle:

${\displaystyle E_{k}={\frac {1}{2}}m_{0}v^{2}+\cdots }$

Without this second term, there would be an additional contribution in the energy when the particle is not moving.

Einstein's view on mass

Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906.^[62][63] In Einstein's first 1905 paper on E = mc², he treated m as what would now be called the rest mass,^[5] and it has been noted that in his later years he did not like the idea of "relativistic mass".^[64]

In older physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass.^[13] Historically, there has been considerable debate over the use of the concept of "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. One view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. Another view, attributed to Norwegian physicist Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.^[65][66]

Einstein's 1905 derivation

Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:

${\displaystyle E_{k}=mc^{2}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)}$.

Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", one of his

Neglecting effects higher than third order in v/c after a Taylor series expansion of the right side of this yields:

${\displaystyle K_{0}-K_{1}={\frac {E}{c^{2}}}{\frac {v^{2}}{2}}.}$

Einstein concluded that the emission reduces the body's mass by E/c², and that the mass of a body is a measure of its energy content.

The correctness of Einstein's 1905 derivation of E = mc² was criticized by German theoretical physicist

In 1911, German physicist Max von Laue gave a more comprehensive proof of M₀ = E₀/c² from the stress–energy tensor,^[78] which was later generalized by German mathematician Felix Klein in 1918.^[79]

Einstein returned to the topic once again after World War II and this time he wrote E = mc² in the title of his article^[80] intended as an explanation for a general reader by analogy.^[81]

Alternative version

An alternative version of Einstein's

${\displaystyle \Delta P={v \over c}{E \over 2c}.}$

The left-moving light carries a little less momentum, by the same amount ΔP. So the total right-momentum in both light pulses is twice ΔP. This is the right-momentum that the object lost.

${\displaystyle 2\Delta P=v{E \over c^{2}}.}$

The momentum of the object in the moving frame after the emission is reduced to this amount:

${\displaystyle P'=Mv-2\Delta P=\left(M-{E \over c^{2}}\right)v.}$

So the change in the object's mass is equal to the total energy lost divided by c². Since any emission of energy can be carried out by a two-step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass.

Radioactivity and nuclear energy

It was quickly noted after the discovery of

Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify them. The theoretical explanation for radioactive decay is given by the nuclear forces responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which was known by then to release enough energy to possibly be "weighed," when missing from the system. However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."^[86]

This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single

Notes

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External links

Wikisource has original text related to this article:

Relativity: The Special and General Theory

Wikimedia Commons has media related to Einstein formula.

• Einstein on the Inertia of Energy – MathPages
• Einstein-on film explaining a mass energy equivalence
• Mass and Energy – Conversations About Science with Theoretical Physicist Matt Strassler
• The Equivalence of Mass and Energy – Entry in the Stanford Encyclopedia of Philosophy
• Merrifield, Michael; Copeland, Ed; Bowley, Roger. "E=mc² – Mass–Energy Equivalence". Sixty Symbols. Brady Haran for the University of Nottingham.